To find the circumcenter of a triangle, we follow these steps:

Steps:

1. Circumcenter Definition:
   The circumcenter is the intersection of the perpendicular bisectors of the sides of a triangle.

2. Midpoints of the Sides:
   Calculate the midpoints of two sides of the triangle.

3. Slopes of the Sides:
   Determine the slopes of the chosen sides, then find the negative reciprocals to get the slopes of the perpendicular bisectors.

4. Equations of the Perpendicular Bisectors:
   Use the midpoint and perpendicular slope to write the equations of the perpendicular bisectors.

5. Intersection of the Perpendicular Bisectors:
   Solve the equations of the two perpendicular bisectors to find their intersection, which is the circumcenter.

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Given Points:

• $A(2, 3)$, $B(-4, -3)$, $C(2, -3)$

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Step 1: Find the midpoints of two sides.

• Midpoint of $AB$:
  $M_{AB} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + (-4)}{2}, \frac{3 + (-3)}{2} \right) = (-1, 0)$

• Midpoint of $AC$:
  $M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 2}{2}, \frac{3 + (-3)}{2} \right) = (2, 0)$

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Step 2: Find the slopes of $AB$ and $AC$.

• Slope of $AB$:
  $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 3}{-4 - 2} = \frac{-6}{-6} = 1$

• Slope of $AC$:
  $m_{AC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 3}{2 - 2} = \text{undefined (vertical line)}$

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Step 3: Find the slopes of the perpendicular bisectors.

• Perpendicular slope of $AB$:
  The negative reciprocal of $1$ is $-1$.

• Perpendicular slope of $AC$:
  Since $AC$ is vertical, its perpendicular slope is $0$ (horizontal line).

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Step 4: Equations of the perpendicular bisectors.

• Perpendicular bisector of $AB$:
  Using point-slope form $y - y_1 = m(x - x_1)$, with slope $-1$ and midpoint $(-1, 0)$:
  $y - 0 = -1(x - (-1)) \implies y = -x - 1$

• Perpendicular bisector of $AC$:
  Using slope $0$ and midpoint $(2, 0)$:
  $y = 0$

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Step 5: Solve for the intersection.

Solve the system of equations:

1. $y = -x - 1$
2. $y = 0$

Substitute $y = 0$ into $y = -x - 1$:
$0 = -x - 1 \implies x = -1$

Thus, the circumcenter is:
$(x, y) = (-1, 0)$

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Final Answer:

The circumcenter is $(-1, 0)$.

Would you like a graphical representation or further details?

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Related Questions for Practice:

1. How is the circumcenter related to the circumcircle of a triangle?
2. Can the circumcenter of a triangle lie outside the triangle? Why?
3. How do you find the circumradius given the circumcenter?
4. What is the circumcenter of a right triangle?
5. How do you prove that the circumcenter is equidistant from all vertices of the triangle?

Tip: The circumcenter is always inside the triangle for acute triangles, on the hypotenuse for right triangles, and outside for obtuse triangles!